1. CHE 185 – PROCESS CONTROL AND DYNAMICS
FREQUENCY RESPONSE ANALYSIS

2. Frequency Response Analysis
Is the response of a process to a sinusoidal input
Considers the effect of the time scale of the input.
Important for understanding the propagation of variability through a process.
Important for terminology of the process control field.
But it is NOT normally used for tuning or design of industrial controllers.

3. Frequency Response Analysis
Process Exposed to a Sinusoidal Input.

4. Frequency Response Analysis
Key Components – INPUT FREQUENCY, ampLitude ratio, phase angle.

5. Frequency Response Analysis
Effect of Frequency on Ar and Φ
PEAK TIME DIFFERENCE Φ= 𝜔 ∆ 𝑡 𝑝 2𝜋 360 °

6. Frequency graphics
Bode plot of Ar and Φ versus frequency ω

7. BODE STABILITY PLOT
BASIS OF BODE PLOT IS A MEASURE OF RELATIVE AMPLITUDE AND PHASE LAG BETWEEN A REGULAR (SINUSOIDAL) SET POINT CHANGE AND THE OUTPUT SIGNAL
THIS TECHNIQUE INDICATES STABILITY OF THE SYSTEM
THE ANALYSIS IS COMPLETED WITH AN OPEN LOOP

8. BODE GENERATION Direct excitation of process.
Ways to Generate Bode Plot INCLUDE:
Direct excitation of process.
Combine transfer function of the process with sinusoidal input.
Substitute s=i w into Gp(s) and convert into real and imaginary components which yield Ar(w) and f(w).
SEE APPLICATION METHOD IN EXAMPLES 11.1 AND 11.2
Apply a pulse test.

9. BODE STABILITY PLOT A SINUSOIDAL SETPOINT IS SENT TO THE LOOP
AFTER THE SYSTEM REACHES STEADY STATE, THERE IS A LAG, CALLED THE PHASE LAG, BETWEEN THE AMPLITUDE PEAK OF THE INLET SIGNAL AND THE AMPLITUDE PEAK OF THE OUTLET SIGNAL
THE FREQUENCY IS ADJUSTED SO THE PHASE LAG OF THE OUTLET SIGNAL IS 180° BEHIND THE INPUT SIGNAL

10. BODE STABILITY PLOT THE RESULTS ARE THEN APPLIED TO A CLOSED LOOP
THE SETPOINT IS CHANGED TO A CONSTANT VALUE
SINCE THE ERROR SIGNAL IS 180° OUT OF PHASE AND IS NEGATIVE RELATIVE TO THE INPUT SIGNAL, IT REINFORCES THE PREVIOUS ERROR SIGNAL

11. BODE STABILITY PLOT
THE AMPLITUDE OF THE ERROR SIGNAL BECOMES THE OTHER FACTOR
IF THE AMPLITUDE OF THE ERROR SIGNAL TO THE AMPLITUDE OF THE ORIGINAL SINUSOID SETPOINT, CALLED THE AMPLITUDE RATIO, IS LESS THAN ONE THEN THE ERROR WILL DECAY TO ZERO OVER TIME,
IF THE AMPLITUDE RATIO WAS EQUAL TO ONE, A PERMANENT STANDING WAVE WILL RESULT
IF THE AMPLITUDE RATIO WAS GREATER THAN ONE, THE ERROR WILL GROW WITHOUT LIMIT.

12. BODE STABILITY PLOT BODE’S STABILITY CRITERION SAYS:
WHEN THE AMPLITUDE RATIO IS LESS THAN ONE, THE SYSTEM IS STABLE
WHEN THE AMPLITUDE RATIO IS GREATER THAN ONE, THE SYSTEM IS UNSTABLE
THE AMPLITUDE RATIO IS DEFINED AS:
WHERE ar REFERS TO AMPLITUDE AS SHOWN IN FIGUREs and IN THE TEXT AND ω IS THE FREQUENCY OF THE SINUSOID

13. BODE STABILITY PLOT BODE PLOTS GENERAL TECHNIQUE TO PLOT
THESE ARE SHOWN FOR FOPDT PROCESSES IN FIGURES AND
A SECOND ORDER PLOT IS FIGURE
GENERAL TECHNIQUE TO PLOT
Write the transfer function in proper form (unit value for lowest order term in denominator)
Separate the transfer function into parts based on poles and zeros
Draw bode diagram for each part
Sum the parts to get the final plot

14. Bode Plot creation example (http://lpsa. swarthmore
Transfer function in proper form 𝐻 𝑠 = 100 𝑠+30 or 𝐻 𝑠 = 𝑠 30 +1
Parts are based on pole at s=30 and constant of 3.3
Pole plot is constant 0 db up to break ω, then drops off
Constant has value of 10.4 db

15. Bode Plot creation example (http://lpsa. swarthmore
Function with real poles and zeros

16. Bode Plot creation example (http://lpsa. swarthmore
Function with real poles and zeros

17. Bode Plot creation example (http://lpsa. swarthmore
Function with pole at origin

18. Bode Plot creation example (http://lpsa. swarthmore
Function with repeated real poles, negative constant

19. Bode Plot creation example (http://lpsa. swarthmore
Function with complex conjugate poles

20. Bode Plot creation example (http://lpsa. swarthmore
Function with multiple poles at origin, complex conjugate zeros

21. Bode Plot creation example (http://lpsa. swarthmore
Function with multiple poles at origin, complex conjugate zeros

22. Bode Plot creation example (http://lpsa. swarthmore
Function with time delay

23. BODE STABILITY PLOT
Developing a Bode Plot from the Transfer Function

24. BODE STABILITY PLOT
Derivation for a First Order Process

25. BODE STABILITY PLOT
Properties of Bode Plots

26. BODE STABILITY PLOT Bode Plot of Complex Transfer Functions
Break transfer function into a product of simple transfer functions.
Identify Ar(ω) and Φ(ω) of each simple transfer function from Table 11.1.
Combine to get Ar(ω) and Φ(ω) for complex transfer function according to properties.
Plot results as a function of ω.

27. BODE STABILITY PLOT BODE PLOTS CAN BE PLOTTED FROM TRANSFER FUNCTIONS
WE CAN SET UP THE TRANSFER FUNCTION: Y(s) – Gp(s)C(s) WHERE C(s) IS THE SINUSOIDAL INPUT
AN INVERSE LaPLACE TRANSFORM OF THE RESULT THEN PROVIDES A TIME FUNCTION

28. BODE STABILITY PLOT
TAKING THIS TO A LIMIT TO ELIMINATE TRANSIENTS THAT WILL DECAY LEAVES THE STANDING WAVE FUNCTION
THIS CAN BE USED TO EVALUATE Ar AS A FUNCTION OF ω
AND φ AS A FUNCTION OF ω
TABLE 11.1 PROVIDES FUNCTIONS TO CALCULATE Ar AND φ FOR A NUMBER OF COMMON TRANSFER FUNCTIONS

29. BODE STABILITY PLOT GAIN MARGIN AND PHASE MARGIN
THE BODE STABILITY CRITERION IS EVALUATED AT THE POINT WHERE φ IS EQUAL TO -180°.
THE FREQUENCY AT THIS POINT IS CALLED THE CRITICAL FREQUENCY

30. BODE STABILITY PLOT GAIN MARGIN AND PHASE MARGIN
THE VALUE OF Ar CALCULATED AT THE CRITICAL FREQUENCY, Ar* DETERMINES THE PROCESS STABILITY
THIS IS EXPRESSED AS THE GAIN MARGIN:
WHEN GM > 1, THE SYSTEM IS STABLE

31. BODE STABILITY PLOT GAIN MARGIN AND PHASE MARGIN
THE PHASE MARGIN IS THE VALUE OF THE PHASE ANGLE AT THE POINT WHERE Ar = 1 AND IS RELATIVE TO THE PHASE ANGLE OF -180°:
(EQUATION )
THE FREQUENCY WHERE THIS CONDITION OCCURS IS CALLED THE CROSSOVER FREQUENCY

32. BODE STABILITY PLOT PULSE TEST
THIS IS AN OPEN LOOP TEST USED TO OBTAIN THE VALUES NECESSARY TO CREATE A BODE PLOT
RESULTS COMPARE THE AMPLITUDE AND THE DURATION TIMES FOR THE INPUT AND OUTPUT VALUES FOR AN OPEN LOOP.
THESE ARE USED WITH EQUATIONS THROUGH TO OBTAIN THE BODE PLOT

33. BODE STABILITY PLOT
PULSE TEST EXAMPLE

34. BODE STABILITY PLOT
Developing a Pulse Test Process Transfer Function

35. BODE STABILITY PLOT
Limitations of Transfer Functions Developed from Pulse Tests
They require an open loop time constant to complete.
Disturbances can corrupt the results.
Bode plots developed from pulse tests tend to be noisy near the crossover frequency which affects GM and PM calculations.

36. NYQUIST DIAGRAM PULSE TEST
COMBINES THE VALUE OF Ar AND φ ON A SINGLE DIAGRAM
OTHERWISE IT HAS NO ADVANTAGE OVER THE BODE PLOTS

37. Closed Loop Frequency Response
REFERENCE FIGURE

38. Example of a Closed Loop Bode Plot
REFERENCE FIGURE

39. Analysis of Closed Loop Bode Plot
REFERENCE FIGURE
At low frequencies, the controller has time to reject the disturbances, i.e., Ar is small.
At high frequencies, the process filters (averages) out the variations and Ar is small.
At intermediate frequencies, the controlled system is most sensitive to disturbances.
The peak frequency indicates the frequency for which a controller is most sensitive.